Quantum Walks with Dynamical Control: Graph Engineering, Initial State Preparation and State Transfer

5.8 MB·

by  Thomas Nitsche

Quantum Walks with Dynamical Control: Graph
Engineering, Initial State Preparation and State
Transfer
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0 Thomas Nitsche1,∗, Fabian Elster1, Jaroslav Novotny´2,
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Aur´el G´abris2,3, Igor Jex2, Sonja Barkhofen1, Christine

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Silberhorn1
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J 1 Applied Physics, University of Paderborn, Warburger Str. 100, 33098 Paderborn,

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Germany
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2Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech

] Technical University in Prague, Bˇrehov´a 7, 11519 Prague, Czech Republic
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3Department of Theoretical Physics, University of Szeged, Tisza Lajos k¨oru´t 84,
p
- H-6720 Szeged, Hungary
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n
E-mail: *: [email protected]
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u
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Abstract. Quantum walks are a well-established model for the study of coherent
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transportphenomenaandprovideauniversalplatforminquantuminformationtheory.

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Dynamically influencing the walker’s evolution gives a high degree of flexibility for
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studyingvariousapplications. Here,wepresenttime-multiplexedfinitequantumwalks
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0 of variable size, the preparation of non-localized input states and their dynamical
2 evolution. As a further application, we implement a state transfer scheme for an
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arbitrary input state to two different output modes. The presented experiments rely
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. on the full dynamical control of a time-multiplexed quantum walk, which includes
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adjustable coin operation as well as the possibility to flexibly configure the underlying
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6 graph structures.
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:
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X PACS numbers: 05.60.Gg
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a
Quantum Walks with Dynamical Control 2
1. Introduction
The well-established concept of quantum walks [1, 2] lays the foundation for the study
of a rich variety of phenomena such as transport dynamics [3, 4, 5, 6], topological phases
[7, 8, 9] or quantum computation [10, 11, 12]. Current experimental implementations
include nuclear magnetic resonance [13, 14], trapped ions [15, 16], atoms [17, 18],
photonicsystems[19,20,21,22,23,24]andwaveguides[25,26,27,28,29,30,31,32,33].
Time-multiplexed quantum walks have been introduced in 2010 with the first
demonstration of a photonic quantum walk on a line [34], which was the first
experimental implementation of a coined discrete quantum walk. Its advantage of
requiring little resources while at the same time offering a high homogeneity and long-
time stability against decoherence due to interferometric stability established a new
experimental platform for studying complex coherence phenomena. By introducing
a fast switching electro-optic modulator the static coin operation was extended to a
dynamical coin in [35], which enabled the investigation of random perturbations with
dynamically changing coins. This experiment succeeded in demonstrating the quantum
to classical crossover by destroying the interferences, and showed Anderson localisation
in a quantum walk system.
In the next step, the concept of time-multiplexed quantum walks was extended to
the second dimension with the first experiment of a quantum walk on a 2-dimensional
lattice [36]. Since this situation is mathematically equivalent to two walkers in one
dimension, this system can be used as a highly controllable simulator for various 2-
particle interactions in one dimension. Still, all of these implementations have been
realized on a static underlying graph and the principle of time-multiplexing appeared
to be incompatible with reconfigurations of the lattice structure. In order to overcome
this restriction, a sophisticated double-step scheme was invented, which allowed the
demonstration of a quantum walk on a dynamical percolation graph [37]. Such a graph
is defined as having its edges probabilistically added or removed in time to imitate
porous or fluctuating media. Using a quantum walk on a dynamical percolation graph,
subtle decoherence effects in presence of randomly distributed, fluctuating gap defects
were studied and clear signatures of non-Markovian behaviour were observed.
In this article, we present three new applications to further emphasize the
capabilities and the flexibility of our current time-multiplexed quantum walk setup:
First, we show the walker’s time evolution on finite graphs with variable sizes. Second,
we exploit the dynamical coin to prepare for the first time in-situ non-localised
initial states with controlled polarization and well-defined phase. By comparing their
evolution, their strong dependency on the input polarization is revealed. And third,
we demonstrate a state transfer scheme as suggested in [38] and route an arbitrary
and unknown initial state to two different detector positions achieving high fidelities
between input and output state up to 99.4%. We find the predicted periodicity in the
time dynamics of the walker localizing at the original position [38].
The article is structured as follows: It starts with a conceptual introduction and
Quantum Walks with Dynamical Control 3
a review of time-multiplexed quantum walks with dynamical coin control in section 2,
including the description of time-multiplexing and the fundamentals on dynamical coin
control. Then we present the experimental implementation of the quantum walk with a
dynamically controlled coin in section 3. Here, we also provide details on experimental
challenges and difficulties inherent in the nature of time-multiplexed systems and give
insights into the realised solutions. These concern the concrete timings, how to handle
the different power levels and how to deal with the ubiquitous losses. Section 4 is
dedicated to the presentation of the experimental results of the three applications of the
dynamical coin: the implementation of a finite walk, in-situ preparation of non-localised
initial states and the demonstration of state transfer schemes. Finally, we conclude in
section 5.
2. Concepts
2.1. Fundamentals
The concept of quantum walks merges the classical model of the random walk with the
quantum mechanical description of a system via its wave function. From the classical
random walk originates the idea that the walker’s path is determined by a coin toss.
Whilethewalkertakesoneofthepossiblepathsinaclassicalrandomwalk,inaquantum
walk the amplitude of the wave function is split up into components taking the different
paths. This results in the superposition of components originating from different paths
and consequently interference effects are observed.
The state |Ψ(cid:105) of the quantum walker is described by a vector in a tensor product space
H ⊗ H of a position Hilbert space H and a coin Hilbert space H . Let |x(cid:105) and |c(cid:105)
x c x c
denote the basis vectors of the position and the coin space then we can write the full
wave function at time t as:
(cid:88)(cid:88)
|Ψ(t)(cid:105) = a (t)|x(cid:105)⊗|c(cid:105) . (1)
x,c
x c
with the time-dependent amplitudes a (t) ∈ C.
x,c
Assuming a time-independent effective Hamiltonian, the evolution of the walker’s
wave function with the number n of discrete time steps is given by the following
expression:
|Ψ(t )(cid:105) = e−in·Hˆ|Ψ(t )(cid:105) . (2)
n 0
ˆ
In the discrete time quantum walk (DTQW), the evolution unitary U for one
ˆ
discrete step consists of a coin operation C, the analog of the classical coin toss, acting
ˆ
only on the internal state and a subsequent step operation S modifying the external
state:
ˆ ˆˆ
|Ψ(t )(cid:105) = U|Ψ(t )(cid:105) = SC |Ψ(t )(cid:105) . (3)
1 0 0
Quantum Walks with Dynamical Control 4
For our realisation of a quantum walk on a 1D grid, the polarisation, expressed in the
basis vectors |H(cid:105) = (1,0)T and |V(cid:105) = (0,1)T, represents the internal state, which then
determines the operation Sˆ on the positions x ∈ Z:
(cid:88)
ˆ
S = (|x+1,H(cid:105)(cid:104)x,H|+|x−1,V(cid:105)(cid:104)x,V|) . (4)
x∈Z
The part of the walker’s wave packet in horizontal polarisation |H(cid:105) undergoes an
increase of position by one, while the part in vertical polarisation |V(cid:105) has the position
ˆ
decreased by one. In general the step operator S carries the information about the
possible paths for the walker, i.e. the pattern of connections (edges) between possible
positions (here: a 1-d line) given by the underlying graph. More complex graph
configurations, e.g. a line with missing links or boundaries, thus require a modified
step operator compared to expression (4).
2.2. Photonic Quantum Walks with Time-Multiplexing
We realise the DTQW with a single photonic walker and use its polarisation as the
coin state. Implementing the quantum walk with the position as the external degree
of freedom, however, limits the scalability of the setup (for the example of a Galton
board-like beam splitter cascade the number of components increases quadratically with
the number of steps). Additionally, decoherence effects will occur due to experimental
inaccuracies of the growing number of components. Thus, we take a different approach
and apply a loop geometry [34]: The concept of time-multiplexing is based on encoding
theexternalstateofthequantumwalkerintothetimedomain(seeFigure1). Initialising
a photonic quantum walk, a coherent laser pulse is sent into the setup with a given input
polarisation. First,thepulsepassesstandardwaveplatesorafastswitchingelectro-optic
modulator (EOM), which realise the desired coin operation (see sec. 2.3 for details).
Then, a polarisation dependent splitting of the pulse is carried out by a polarising beam
splitter (PBS). By routing the pulses afterwards through single-mode fibres (SMF) of
different lengths we introduce a well-defined time delay between them. This splitting
operationincombinationwiththetimedelayandthesubsequentmergingofthepathsat
the second PBS constitutes one shift of the time-multiplexed quantum walk according
to eq. 4. We interpret the part arriving earlier as having undergone a reduction of
the position by one and the component arriving later as having been subjected to an
increase of the position by one. Hence, every spatial position is uniquely represented by
its arrival time.
As we want to achieve the n-times application of coin and the step operation, the
outputisagainfedintothesetup,resultinginaloop-likearchitectureoftheexperimental
implementation. In every step a small amount of intensity is coupled out and sent to
the detection unit consisting of another PBS and two avalanche photodiodes (APDs)
allowing for polarisation resolved measurements of the walker’s time evolution.
Figure 2 illustrates the 3 relevant timings in the time-multiplexed quantum walk: The
Quantum Walks with Dynamical Control 5
Figure 1. Schematicofourimplementationofatime-multiplexedquantumwalkwith
an electro-optic modulator (EOM) and a half-wave plate (HWP) (here exemplarily)
carrying out the coin operation, two single mode fibres (SMFs) introducing the time
delay and the avalanche photo-diodes (APDs) used for the read-out of the walker’s
state
splitting of the two polarisation components introduces a delay of τ between them
Pos
given by the length difference between the two SMFs. The roundtrip time τ is defined
RT
by the length of the shorter fibre (plus the free-space parts) and denotes the step
separation. Two subsequent experiments are delayed by τ which must be set long
Rep
enough to observe the desired number of steps without overlapping.
The proper tailoring of these timings is essential for the time-multiplexing scheme and
we dedicate sec. 3.1 to its experimental details.
2.3. Dynamical coin and dynamical graph
As described above a quantum walk includes a coin operation acting on the internal
degree of freedom, in our case the polarisation. In the experiment the polarisation
can be statically rotated in a straight-forward way by applying retardation plates. For
example a typical coin, implemented by a half-wave plate (HWP) at an angle of 22.5◦,
is the Hadamard coin, which transforms an input comprising one polarization into a
fair superposition of both polarizations. Another balanced coin can be implemented
by a quarter-wave plate (QWP) at an angle of 45◦ (see subsection 2.4). Quantum
walks having the spatial position as the external degree of freedom can rely on static
phase-shifters in order to implement a position-dependent coin operation [39]. In a
time-multiplexed setup in a loop architecture, however, such devices can only realise
Quantum Walks with Dynamical Control 6
Figure 2. Schematic illustration of the 3 relevant timings (not to scale): the position
spacing τ , the roundtrip spacing τ and the time τ between two consecutive
Pos RT Rep
experiments defining the repetition rate; counts for vertically polarized light are
represented by blue bars and counts for horizontally polarized light by orange bars
(schematically)
operations which are the same for all steps and positions.
Introducing a dynamical coin dramatically increases the versatility of a quantum walk
as a model system: It offers new possibilities, e.g. in altering the structure of the
underlying graph [37] (see also subsection 4.1) as well as in “in-situ” preparation of the
initial state of the quantum walk (see subsection 4.2).
A direct manipulation of the graph structure would mean to modify the step operator
which carries the information on the walker’s possible paths. Since in our setup the
step operation is realised by a fixed combination of PBSs and SMFs this can obviously
not be dynamically changed. However, the displacement direction in the step operation
depends on the polarisation (see equation 4) and the polarization can be addressed
dynamically via fast switching EOMs. Thus a clever manipulation of the polarization
allows to prevent that intensity from a certain position is shifted in a certain direction
(e.g. to the right), which is the same effect as breaking the respective link (in this case
the right one) would have.
We will present here in more detail how the dynamical coin is harnessed for the
implementation of a quantum walk on a finite graph: Figure 3 shows in a Galton
board-like scheme a possible evolution of the walker using two specific coin operations,
ˆ
the reflection and the transmission. A reflection operation R (grey box) denotes a
polarization switching from |H(cid:105) to |V(cid:105) and vice versa. Leaving the polarisations
ˆ
unchanged corresponds to the transmission operation T (white box). In the given
example, we intend to limit the spread of the walker’s external state to the positions
−1,0,1, thus it must not evolve into the red areas. This corresponds to a reflection
operator at x = −1 for the left boundary and at x = 1 for the right boundary.
Quantum Walks with Dynamical Control 7
Figure 3. Illustration of how the spread of the walker’s external state is limited
via polarisation switching; a reflection Rˆ is represented by a grey box and switches
the horizontal (red lines) into vertical (blue lines) polarization and vice versa. The
transmission operation Tˆ is represented by a white box leaving the polarizations
unchanged. Note that in the setup the spatial position is in the experiment mapped
into the time domain.
The example illustrates on the one hand that time-multiplexing techniques in
combination with dynamical coin control provide a powerful tool for implementing not
onlydynamicalcoinoperationsbutalsonon-trivialgraphstructures. Ontheotherhand,
it also shows the significance of properly timed polarisation switchings, which require
accurate engineering of the hardware and the software of the setup. The presentation
of its details will be given in the following sections.
2.4. Implementation of dynamical coin operations via an EOM
For the implementation of non-trivial graphs a device is needed that switches between
ˆ ˆ ˆ
reflection and transmission operations R, T and, possibly, a third mixing coin C. In
order to address single positions the device must exhibit a switching speed that is
comparable to the position separation τ (here: 46.5ns).
Pos
In order to meet the challenging requirements on homogeneity, accuracy of rotation
angle and switching speed, we thus apply an electro-optic modulator (EOM, see Figure
4). This device operates based on the Pockels effect to manipulate the polarization of
lightpassingthroughit: Therefractiveindexofamediumischangedalongthedirection
of an applied voltage, thus introducing or altering birefringence, which can be done at
high speeds. The Pockels cell used in the setup consists of a rubidium titanyl phosphate
(RTP) crystal that exhibits natural birefringence. Since we need to set the actually
realised operation by switching on or off the external voltage, the natural birefringence
has to be compensated for: This was achieved by cutting the crystal in two halves and
rotate them against each other by 90◦.
In order to obtain the desired reflection and transmission matrices, aligning the
Quantum Walks with Dynamical Control 8
Figure 4. Schematic of the two EOM crystals; the reference frame is given by the
orientation of the crystal axes and is rotated by 45◦ with respect to |H(cid:105) and |V(cid:105): the
phase φ originating from the natural birefringence is accumulated along the x-axes,
while the phase Φ originating from the application of E(cid:126) is accumulated along the
U
z-axes of the crystals.
crystal to an rotation angle of 45◦ with respect to the H- and V-axes is crucial. Still it is
not trivial which concrete operators can be realised with this device, since even phases
are significant because the interference patterns are strongly phase-sensitive. As the
external voltage is applied along the z-axes of the crystals we can derive the expression
for the rotation operation of the EOM in the H-V-basis:
Cˆ (U) = R(45◦)·Cˆ ·Cˆ ·R(−45◦)
EOM crystal 1 crystal 2
(cid:32) (cid:33)(cid:32) (cid:33)(cid:32) (cid:33) (cid:32) (cid:33)
1 1 −1 eiΦU 0 eiϕ 0 1 1 1
= √ √ (5)
2 1 1 0 eiϕ 0 e−iΦU 2 −1 1
(cid:32) (cid:33)
cos(Φ ) isin(Φ )
= eiϕ · U U .
isin(Φ ) cos(Φ )
U U
where R(±45◦) are the rotation matrices for the basis transform, Cˆ and Cˆ
crystal 1 crystal 2
represent the operation in the respective halves of the crystal and Φ the additional
U
phase induced by the Pockels effect. In principle all operators of this shape can be
realised, just by setting an appropriate voltage to attain the needed phase Φ . However,
U
the entries of the matrices either assume merely real or imaginary values, so that the
possible operations do not exhaust the full space of unitary 2×2 matrices and we have
to stay within this constraint regarding the phases when choosing coin operations. In
the coherent evolution of the walker these phase terms are of fundamental significance.
For the modification of the graph structure we are interested in 3 special cases:
ˆ
For Φ = 0 the EOM realises the transmission operator T = 1 when setting the global
U
phase ϕ = 0. For an appropriate choice of U yielding Φ = π we obtain the reflection
U 2
operator Rˆ with i-phases on the off-diagonals. Setting Φ to π realises a balanced coin
U 4
Quantum Walks with Dynamical Control 9
Figure 5. Bloch sphere representation of the EOM switching states corresponding to
the three operations Rˆ, Cˆ and Tˆ exemplary for an horizontally polarised input
QWP
state; a static QWP is placed in front of the EOM to realise Cˆ in case no voltage
QWP
is applied
ˆ
C that transforms e.g. |H(cid:105) into an equal superposition of both polarisations and is
QWP
equivalent to a quarter-wave plate (QWP):
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
1 0 0 i 1 1 i
Tˆ = , Rˆ = , Cˆ = √ . (6)
QWP
0 1 i 0 2 i 1
For our EOM in use there is another restriction: Within one experiment it is able
to switch on or off a manually preset voltage with positive or negative sign, that means
that we are limited to the three phases Φ = 0,±Φ for one experiment. In order to
U U0
ˆ
implement the three different operations anyhow, we place a static QWP with C in
QWP
front of it. Now we have a balanced coin when no voltage is applied in the EOM and
can switch to reflection or transmission by applying a positive or negative voltage U ,
0
respectively (Fig. 5).
3. Experimental implementation
3.1. Timings
The time-multiplexing scheme is based on three different time scales which have to be
designed appropriately in order to guarantee the unique mapping between arrival time
and step and position assignment of a pulse. In addition, technical restrictions must be
taken into account, e.g the dead times of the detectors, the overall measurement time
and where required the switching speed for the dynamical coin.
The smallest time scale is the position separation τ given by the runtime
Pos
difference in the two fibres. Since the walker’s wave function is distributed over more
and more positions with growing step number, the roundtrip time τ must be chosen
RT
long enough to prevent the overlapping of pulses belonging to different steps up to a
maximal step m. Such an unambiguous assignment of the positions requires that the
latest position of the n-th step has no overlap with the earliest position of the (n+1)-th
Quantum Walks with Dynamical Control 10
step (see Figure 2). Since the number of possible positions of the m-th step equals m+1,
we find the following relation between the time scales:
(m+1)·τ < τ (7)
Pos RT
In the presented setup we chose fibres of lengths 145m and 135m, respectively. On
the one hand, the resulting length difference of 10m leads to a position separation of
τ = 46.5ns matching the deadtimes of the APDs and the switching speed of the EOM
Pos
of around 50ns. On the other hand, the spacing of the roundtrips τ is determined by
RT
the length of the shorter fibre (plus the free space parts), which corresponds to 685ns
allowing for adding up 14 multiples of τ . Thus, our setup allows for more than 13
Pos
steps only if explicit boundary conditions apply, restricting the walk to a finite number
of positions.
The duration of one experimental run τ is then given by the maximal step
Rep
number times the roundtrip time. In order to obtain reliable statistics for ensemble
measurements, the experimental data is obtained via averaging over many runs of the
experiment taking place with a repetition rate of f = τ−1 . Here, the entire duration
Rep Rep
of one experimental run of nearly 9µs allows for a repetition rate of 110kHz.
For the finite walk (see sec. 4.1) the limited number of populated positions
prevents their overlapping, thus we can achieve step numbers up to 21 while running
the experiment at 50kHz. In both cases we get reliable statistics of the experimental
data within a few minutes.
3.2. Analysis of Photon Numbers for Reliable Detection
Forseveralapplicationsofquantumwalksweaimforareliablemonitoringofthewalker’s
full time evolution and not only for a final outcome distribution. For achieving this goal
we have to deal with several experimental challenges, e.g. a broad power range of pulses
arriving at the detectors, reduction of multi-photon contributions at the APDs and
accepting higher round trip losses when sending light to the detection unit in every
step. Hence a proper analysis of power levels and photon numbers is essential.
In order to track the walker’s dynamics over time, a certain proportion of the light
is coupled out in every step via probabilistic outcouplers and sent to the detetion unit.
Similarly, the photons are initially fed into the setup via a probabilistic incoupler. The
proper adjustment of the initial power level as well as the in- and outcoupling ratio, is
criticalforreliablymeasuringthewalker’sstateforstepnumbersashighaspossible: On
the one hand, the intrinsic properties of “click”-detectors demand for negligible multi-
photoncontributionsinthestepstobeanalyzed. Ontheotherhand, wewanttoconduct
measurements with an acceptable signal-to-noise ratio (SNR) for as high step numbers
n as possible. However, due to the loop architecture the losses, including the outcoupled
portions of the light, scale exponentially with n limiting the maximal observable step
number(foradetailedanalysisoftheoccuringlossesseenextsubsection). Consequently,
a trade-off between a sufficient SNR for higher steps and a low enough initial energy to
not damage the detectors and to reduce multi-photon contributions must be found.